\documentclass[a4paper,reqno]{amsart} %\usepackage[notcite,notref]{showkeys} \usepackage[neverdecrease]{paralist} \usepackage{footmisc} \newcommand{\included}{\subseteq} \newcommand{\pincluded}{\subset} \newcommand{\nincluded}{\nsubseteq} \newcommand{\includes}{\supseteq} \usepackage{mathrsfs} \newcommand{\sig}{\mathscr S} \usepackage{upgreek} \newtheorem{theorem}{Theorem} \newtheorem*{lemma}{Lemma} \newcommand{\str}[1]{\mathfrak{#1}} % structure with universe #1 \newcommand{\diag}[1]{\operatorname{diag}(#1)} \usepackage{amssymb} % for \restriction \renewcommand{\restriction}{\mathbin{\upharpoonright}} \newcommand{\elsub}{\preccurlyeq} \newcommand{\Forall}[1]{\forall{#1}\;} \newcommand{\Exists}[1]{\exists{#1}\;} %\newcommand{\lto}{\Rightarrow} \newcommand{\lto}{\to} \newcommand{\stnd}[1]{\mathbb{#1}} \newcommand{\Z}{\stnd Z} \newcommand{\Q}{\stnd Q} \newcommand{\F}{\stnd F} \usepackage{bm} \newcommand{\thy}[1]{\mathrm{#1}} % for the following: %\newcommand{\ACF}{\thy{ACF}} % algebraically closed fields %\newcommand{\SCF}{\thy{SCF}} % separably closed fields \newcommand{\DF}{\thy{DF}} % differential fields %\newcommand{\DPF}{\thy{DPF}} % differentially perfect fields \newcommand{\ACFA}{\thy{ACFA}} % fields with generic autom. \newcommand{\mDF}{\text{$m$-$\thy{DF}$}} % fields with m derivations \newcommand{\mpDF}{\text{$(m+1)$-$\thy{DF}$}} % fields with m+1 derivations \newcommand{\oDF}{\text{$\upomega$-$\thy{DF}$}} % fields with omega derivations %\newcommand{\mDPF}{\text{$m$-$\thy{DPF}$}} % diff. perf. fields with m derivations \newcommand{\DCF}{\thy{DCF}} % differentially closed fields \newcommand{\mDCF}{\text{$m$-$\thy{DCF}$}} % differentially closed fields % with m derivations \newcommand{\oDCF}{\text{$\upomega$-$\thy{DCF}$}} % differentially closed fields % with omega-many derivations \newcommand{\mpDCF}{\text{$(m+1)$-$\thy{DCF}$}} % differentially closed fields % with m+1 derivations \newcommand{\VSp}[1]{\operatorname{VS}_{#1}} % vector-spaces with an % n-ary predicate for linear dependence \newcommand{\VSpm}[1]{\operatorname{VS}_{#1}^{\mathrm m}} % vector-spaces %of minumum dimension n with an n-ary predicate for linear dependence \newcommand{\VSpr}[1]{\operatorname{VS}_{#1}^{\mathrm r}} % reducts of % models of this \newcommand{\VSpst}[1]{{\VSp{#1}}\!^*} % model-companion for these \newcommand{\Mod}[2][\included]{\operatorname{Mod}^{#1}(#2)} %\newcommand{\rem}[1]{\marginpar{\texttt{#1}}} %\newcommand{\ut}[1][T]{\bigcup_{n\in\upomega}{#1}_n} \renewcommand{\theequation}{\fnsymbol{equation}} \renewcommand{\phi}{\varphi} \renewcommand{\setminus}{\smallsetminus} \renewcommand{\leq}{\leqslant} \renewcommand{\geq}{\geqslant} \renewcommand{\vec}[1]{\bm{#1}} \newcommand{\alg}{^{\mathrm{alg}}} \usepackage{url} \begin{document} \title{Chains of Theories and Companionability} \author[\"O. Kasal]{\"Ozcan Kasal} \address{Middle East Technical University, Northern Cyprus Campus} \author[D. Pierce]{David Pierce} \address{Mimar Sinan Fine Arts University, Istanbul} \email{dpierce@msgsu.edu.tr} \date{\today} \begin{abstract} The theory of fields with a countably infinite family of commuting derivations is not companionable; but if the axiom is added whereby the characteristic of the fields is zero, then the resulting theory is companionable. Each of these two theories is the union of a chain of companionable theories. In a signature with predicates, in all finite numbers of arguments, for linear dependence of vectors, the two-sorted theory of vector-spaces with their scalar-fields is companionable, and it is the union of a chain of companionable theories, but its model-companion is not the union of the model-companions of the theories in the chain. Finally, the union of a chain of non-companionable theories may be companionable. \end{abstract} \maketitle A \textbf{theory} in a given signature is a set of sentences, in the first-order logic of that signature, that is closed under logical implication. We shall consider chains $(T_m\colon m\in\upomega)$ of theories: this means \begin{equation}\label{eqn:no*} T_0\included T_1\included T_2\included\dotsb \end{equation} The signature of $T_m$ will be $\sig_m$, so automatically $\sig_0\included\sig_1\included\sig_2\included\dotsb$% %\phantom{. }Let us denote the theory $\bigcup_{k\in\upomega}T_k$ by $U$. In one motivating example, $\sig_m$ is $\{0,1,-,+,{}\cdot{},\partial_0,\dots,\partial_{m-1}\}$, the signature of fields with $m$ additional singulary function-symbols; and $T_m$ is $\mDF$, the theory of fields (of any characteristic) with $m$ commuting derivations. In this example, each $T_{m+1}$ is a \textbf{conservative extension} of $T_m$, that is, $T_{m+1}\includes T_m$ and every sentence in $T_{m+1}$ of signature $\sig_m$ is already in $T_m$. We establish this by showing that every model of $T_m$ expands to a model of $T_{m+1}$. (This condition is sufficient, but not necessary \cite[\S2.6, exer.~8, p.~66]{MR94e:03002}.) If $(K,\partial_0,\dots,\partial_{m-1})\models\mDF$, then $(K,\partial_0,\dots,\partial_m)\models\mpDF$, where $\partial_m$ is the $0$-derivation. The union of the theories $\mDF$ can be denoted by $\oDF$: it is the theory of fields with $\upomega$-many commuting derivations. Each of the theories $\mDF$ has a \emph{model-companion,} called $\mDCF$ \cite{2007arXiv0708.2769P}; but we shall show (as Theorem~\ref{thm:oDF} below) that $\oDF$ has no model-companion. Let us recall that a \textbf{model-companion} of a theory $T$ is a theory $T^*$ in the same signature such that \begin{inparaenum} \item $T_{\forall}=T^*{}_{\forall}$, that is, every model of one of the theories embeds in a model of the other, and \item $T^*$ is \textbf{model-complete,} that is, $T^*\cup\diag{\str M}$ axiomatizes a complete theory for all models $\str M$ of $T^*$. \end{inparaenum} Here $\diag{\str M}$ is the quantifier-free theory of $\str M$ with parameters: equivalently, $\diag{\str M}$ is the theory of all structures in which $\str M_M$ embeds. (These notions, with historical references, are reviewed further in \cite{2007arXiv0708.2769P}.) A theory has at most one model-companion, by an argument with interwoven elementary chains. Let $\mDF_0$ be $\mDF$ with the additional requirement that the field have characteristic $0$. Then $\mDF_0$ has a model-companion, called $\mDCF_0$ \cite{MR2001h:03066}. We shall show (as Theorem~\ref{thm:oDF0} below) that $\mDCF_0\included\mpDCF_0$. It will follow then that the union $\oDF_0$ of the $\mDF_0$ has a model-companion, which is the union of the $\mDCF_0$. This is by the following general result, which has been observed also by Alice Medvedev~\cite{Medvedev,Medvedev-preprint}. Again, the theories $T_k$ are as in \eqref{eqn:no*} above. \begin{theorem}\label{thm:1} Suppose each theory $T_k$ has a model-companion $T_k{}^*$, and \begin{equation}\label{eqn:*} T_0{}^*\included T_1{}^*\included T_2{}^*\included\dotsb \end{equation} Then the theory $\bigcup_{k\in\upomega}T_k$ has a model-companion, namely $\bigcup_{k\in\upomega}T_k{}^*$. \end{theorem} \begin{proof} Write $U$ for $\bigcup_{k\in\upomega}T_k$, and $U^*$ for $\bigcup_{k\in\upomega}T_k{}^*$. Suppose $\str A\models U$, and $\Gamma$ is a finite subset of $U^*\cup\diag{\str A}$. Then $\Gamma$ is a subset of $T_k{}^*\cup\diag{\str A\restriction\sig_k}$ for some $k$ in $\upomega$, and also $\str A\restriction\sig_k\models T_k$. Since $(T_k{}^*)_{\forall}\included T_k$, we conclude that $\Gamma$ is consistent. Therefore $U^*\cup\diag{\str A}$ is consistent. Thus $U^*{}_{\forall}\included U$. By symmetry $U_{\forall}\included U^*$. Similarly, if $\str B\models U^*$, then $T_k{}^*\cup\diag{\str B\restriction\sig_k}$ axiomatizes a complete theory in each case, and therefore $U^*\cup\diag{\str B}$ is complete. \end{proof} The foregoing proof does not require that the signatures $\sig_k$ form a chain, but needs only that every finite subset of $\bigcup_{k\in\upomega}\sig_k$ be included in some $\sig_k$. This is the setting for Medvedev's \cite[Prop.~2.4, p.~6]{Medvedev-preprint}, which then has the same proof as the foregoing. Also in Medvedev's setting, each $T_{k+1}{}^*$ is a conservative extension of $T_k{}^*$; but only the weaker assumption $T_k{}^*\included T_{k+1}{}^*$ is needed in the proof. Medvedev notes that many properties that the theories $T_k$ might have are `local' and are therefore preserved in $\bigcup_{k\in\upomega}T_k$: examples are completeness, elimination of quantifiers, stability, and simplicity. In her main application, $\sig_n$ is the signature of fields with singulary operation-symbols $\sigma_{m/n!}$, where $m\in\Z$; and $T_n$ is the theory of fields on which the $\sigma_{m/n!}$ are automorphisms such that \begin{equation*} \sigma_{k/n!}\circ\sigma_{m/n!}=\sigma_{(k+m)/n!}. \end{equation*} Then $T_n$ includes the theory $S_n$ of fields with the single automorphism $\sigma_{1/n!}$. Using \cite[\S1]{MR2505433} (which is based on \cite[ch.~5]{MR94e:03002}), we may observe at this point that reduction of models of $T_n$ to models of $S_n$ is actually an equivalence of the categories $\Mod{T_n}$ and $\Mod{S_n}$, whose objects are models of the indicated theories, and whose morphisms are embeddings. We thus have at hand a (rather simple) instance of the hypothesis of the following theorem. \begin{theorem}\label{thm:bi} Suppose $(I,J)$ is a bi-interpretation of theories $S$ and $T$ such that $I$ is an equivalence of the categories $\Mod S$ and $\Mod T$. If $S$ has the model-companion $S^*$, and $S\included S^*$, then $T$ also has a model-companion, which is the theory of those models $\str B$ of $T$ such that $J(\str B)\models S^*$. \end{theorem} \begin{proof} The class of models $\str B$ of $T$ such that $J(\str B)\models S^*$ is elementary. Let $T^*$ be its theory. Then $T\included T^*$. Suppose $\str B\models T$. Then $J(\str B)\models S$, so $J(\str B)$ embeds in a model $\str A$ of $S^*$. Consequently $I(J(\str B))$ embeds in $I(\str A)$. Also $I(\str A)\models T^*$, since $\str A\cong J(I(\str A))$. Since also $\str B\cong I(J(\str B))$, we conclude that $\str B$ embeds in a model of $T^*$. Finally, $T^*$ is model-complete. Indeed, suppose now $\str B$ and $\str C$ are models of $T^*$ such that $\str B\included\str C$. Then $J(\str B)$ embeds in $J(\str C)$, and these structures are models of $S^*$, so the embedding is elementary. Therefore an elementary embedding of $I(J(\str B))$ in $I(J(\str C))$ is induced. By the equivalence of the categories, $\str B\preccurlyeq\str C$. \end{proof} In the present situation, the theory $S_n$ has a model-companion \cite{MR99c:03046,MR2000f:03109}; let us denote this by $\ACFA_n$. By the theorem then, $T_n$ has a model-companion $T_n{}^*$, which is axiomatized by $T_n\cup\ACFA_n$. We have $\ACFA_n\included T_{n+1}{}^*$ by \cite[1.12, Cor.~1, p.~3013]{MR2000f:03109}. By Theorem~\ref{thm:1} then, $\bigcup_{n\in\upomega}T_n$ has a model-companion, which is the union of the $T_n{}^*$. Medvedev calls this union $\Q\ACFA$; she shows for example that it preserves the simplicity of the $\ACFA_n$, as noted above, though it does not preserve their supersimplicity. The following is similar to the result that the theory of fields with a derivation \emph{and} an automorphism (of the field-structure only) has no model-companion \cite{MR2114160}. The obstruction lies in positive characteristics $p$, where all derivatives of elements with $p$-th roots must be $0$. \begin{theorem}\label{thm:oDF} The theory $\oDF$ has no model-companion. \end{theorem} \begin{proof} We use that an $\forall\exists$ theory $T$ has a model-companion if and only if the class of its \emph{existentially closed} models is elementary, and in this case the model-companion is the theory of this class \cite{MR0277372}. (A model $\str A$ of $T$ is an \textbf{existentially closed} model, provided that if $\str B\models T$ and $\str A\included\str B$, then $\str A\preccurlyeq_1\str B$, that is, all quantifier-free formulas over $A$ that are soluble in $\str B$ are soluble in $\str A$.) For each $n$ in $\upomega$, the theory $\oDF$ has an existentially closed model $\str A_n$, whose underlying field includes $\mathbb F_p(\alpha)$, where $\alpha$ is transcendental; and in this model, \begin{equation*} \partial_k\alpha= \begin{cases} 1,&\text{ if }k=n,\\ 0,&\text{ otherwise.} \end{cases} \end{equation*} Then $\alpha$ has no $p$-th root in $\str A_n$. Therefore, in a non-principal ultraproduct of the $\str A_n$, $\alpha$ has no $p$-th root, although $\partial_n\alpha=0$ for all $n$ in $\upomega$, so that $\alpha$ does have a $p$-th root in some extension. Thus the ultraproduct is not an existentially closed model of $\oDF$. Therefore the class of existentially closed models of $\oDF$ is not elementary. \end{proof} It follows then by Theorem~\ref{thm:1} that $\mDCF\nincluded\mpDCF$ for at least one $m$. We could contrive examples to show this independently; but this by itself would not be enough to establish the last theorem. For, by the results of \cite{MR2505433}, it is possible for each $T_k$ to have a model-companion $T_k{}^*$, while $\bigcup_{k\in\upomega}T_k$ has a model-companion that is not $\bigcup_{k\in\upomega}T_k{}^*$. We may even require $T_{k+1}$ to be a conservative extension of $T_k$. Indeed, if $k>0$, then in the notation of \cite{MR2505433}, $\VSp k$ is the theory of vector-spaces with their scalar-fields in the signature $\{+,-,\bm0,\circ,0,1,*,P^k\}$, where $\circ$ is multiplication of scalars, and $*$ is the action of the scalar-field on the vector-space, and $P^k$ is $k$-ary linear dependence. In particular, $P^2$ may written also as $\parallel$. Then $\VSp k$ has a model-companion, $\VSpst k$, which is the theory of $k$-dimensional vector-spaces over algebraically closed fields \cite[Thm~2.3]{MR2505433}. Let $\VSp{\upomega}=\bigcup_{1\leq k<\upomega}\VSp k$. (This was called $\VSp{\infty}$ in \cite{MR2505433}.) This theory has the model-companion $\VSpst{\upomega}$, which is the theory of infinite-dimensional vector-spaces over algebraically closed fields \cite[Thm~2.4]{MR2505433}. \begin{theorem}\label{thm:vs} If $1\leq n<\upomega$, let \begin{equation*} T_n=\bigcup_{1\leq k\leq n}\VSp k. \end{equation*} Then $T_n$ has a model-companion $T_n{}^*$, which is axiomatized by $\VSpst n\cup T_n$. Also $T_{n+1}$ is a conservative extension of $T_n$. However, the model-companion $\VSpst{\upomega}$ of the union $\VSp{\upomega}$ of the chain $(T_n\colon 1\leq n<\upomega)$ is not the union of the $T_n{}^*$. \end{theorem} \begin{proof} Every vector-space can be considered as a model of any $\VSp k$ and hence of any $T_k$. In particular, $T_{n+1}$ is a conservative extension of $T_n$. The relation $P^1$ is defined by $\vec x=\vec0$. Let $\VSpm n$ be axiomatized by $\VSp n$ and the requirement that the space have dimension at least $n$. If $n>2$, then there are existential formulas that, in each model of $\VSpm n$, define the relation $\parallel$ and its complement \cite[\S2, p.~431]{MR2505433}. Similarly, if $2\leq k1$, then reduction of $\VSp n$ to $\VSpr n$ is an equivalence of categories $\Mod{\VSp n}$ and $\Mod{\VSpr n}$, and therefore $\VSpr n$ is companionable. (This was mentioned at \cite[p.~431]{MR2505433}, but the details, and in particular Theorem~\ref{thm:bi}, were not spelled out.) So there is an alternative, one-sorted version of the last theorem, where $T_n$ is $\bigcup_{1\leq k\leq n}\VSpr k$. The implication $\ref{ext}\Rightarrow\ref{red}$ in the following is used implicitly at \cite[1.12, p.~3013]{MR2000f:03109} to establish the result used above, that if $(K,\sigma)$ is a model of $\ACFA$, then so is $(K,\sigma^m)$, assuming $m\geqslant 1$. \begin{theorem}\label{thm:cond} Assuming as usual $T_0\included T_1$, where each $T_k$ has signature $\sig_k$, we consider the following conditions. \begin{compactenum}[$A$.] \item\label{ext} For every model $\str A$ of $T_1$ and model $\str B$ of $T_0$ such that \begin{equation}\label{eqn:B} \str A\restriction\sig_0\included\str B, \end{equation} there is a model $\str C$ of $T_1$ such that \begin{align}\label{eqn:ABC} \str A&\included\str C,& \str B&\included\str C\restriction\sig_0. \end{align} \item\label{red} The reduct to $\sig_0$ of every existentially closed model of $T_1$ is an existentially closed model of $T_0$. \item\label{AP} $T_0$ has the \emph{Amalgamation Property:} if one model embeds in two others, then those two in turn embed in a fourth model, compatibly with the original embeddings. \item\label{ec} $T_1$ is $\forall\exists$ (so that every model embeds in an existentially closed model). \end{compactenum} We have the two implications \begin{align*} \ref{ext}&\implies\ref{red},& \ref{red}\And\ref{AP}\And\ref{ec}&\implies\ref{ext}, \end{align*} but there is no implication among the four conditions that does not follow from these. This is true, even if $T_1$ is required to be a conservative extension of $T_0$. \end{theorem} \begin{proof} Suppose $\ref{ext}$ holds. Let $\str A$ be an existentially closed model of $T_1$, and let $\str B$ be an arbitrary model of $T_0$ such that \eqref{eqn:B} holds. By hypothesis, there is a model $\str C$ of $T_1$ such that \eqref{eqn:ABC} holds. Then $\str A\preccurlyeq_1\str C$, and therefore $\str A\restriction\sig_0\preccurlyeq_1\str C\restriction\sig_0$, and \emph{a fortiori} $\str A\restriction\sig_0\preccurlyeq_1\str B$. Therefore $\str A\restriction\sig_0$ must be an existentially closed model of $T_0$. Thus $\ref{red}$ holds. Suppose conversely $\ref{red}$ holds, along with $\ref{AP}$ and $\ref{ec}$. Let $\str A\models T_1$ and $\str B\models T_0$ such that \eqref{eqn:B} holds. We establish the consistency of $T_1\cup\diag{\str A}\cup\diag{\str B}$. It is enough to show the consistency of \begin{equation}\label{eqn:T} T_1\cup\diag{\str A}\cup\{\Exists{\vec x}\phi(\vec x)\}, \end{equation} where $\phi$ is an arbitrary quantifier-free formula of $\sig_0(A)$ that is soluble in $\str B$. By $\ref{ec}$, there is an existentially closed model $\str C$ of $T_1$ that extends $\str A$. By $\ref{red}$ then, $\str C\restriction\sig_0$ is an existentially closed model of $T_0$ that extends $\str A\restriction\sig_0$. By $\ref{AP}$, both $\str B$ and $\str C\restriction\sig_0$ embed over $\str A\restriction\sig_0$ in a model of $T_0$. In particular, $\phi$ will be soluble in this model. Therefore $\phi$ is already soluble in $\str C\restriction\sig_0$ itself. Thus $\str C$ is a model of \eqref{eqn:T}. Therefore $\ref{ext}$ holds. The foregoing arguments eliminate the five possibilities marked $X$ on the table below, \begin{table}[h] \begin{equation*} \begin{array}{r|*{16}{c}} &\text{\ref{1}}&X&\text{\ref{2}}&\text{\ref{3}}&\text{\ref{4}}&X&\text{\ref{5}}&\text{\ref{6}}&\text{\ref{7}}&X&\text{\ref{8}}&\text{\ref{9}}&\text{\ref{10}}&X&X&\text{\ref{11}}\\\hline \ref{ext}&0&1&0&1&0&1&0& 1&0&1&0& 1&0&1&0& 1\\ \ref{red}&0&0&1&1&0&0&1& 1&0&0&1& 1&0&0&1& 1\\ \ref{AP}&0&0&0&0&1&1&1& 1&0&0&0& 0&1&1&1& 1\\ \ref{ec}&0&0&0&0&0&0&0& 0&1&1&1& 1&1&1&1& 1 \end{array} \end{equation*} \end{table} where $0$ means false, and $1$, true. We give examples of each of the remaining cases, numbered according to the table. In each example, $T_0$ will be the reduct of $T_1$ to $\sig_0$. We shall denote by $\sig_{\mathrm f}$ the signature $\{+,{}\cdot{},-,0,1\}$ of fields; and by $\sig_{\mathrm{vs}}$, the signature $\{+,-,\bm0,\circ,0,1,*\}$ of vector-spaces as two-sorted structures. \begin{asparaenum}[1.] \item\label{1} We first give an example in which none of the four lettered conditions hold. Let $\sig_0=\sig_{\mathrm f}\cup\{a,b\}$ and $\sig_1=\sig_0\cup\{c\}$. Let $T_1$ be the theory of fields of characteristic $p$ with distinguished elements $a$, $b$, and $c$ such that $\{a,c\}$ or $\{b,c\}$ is $p$-independent, and if $\{b,c\}$ is $p$-independent, then so is $\{b,c,d\}$ for some $d$. Then $T_0$ is the theory of fields of characteristic $p$ in which, for some $c$, $\{a,c\}$ or $\{b,c\}$ is $p$-independent, and if $\{b,c\}$ is $p$-independent, then so is $\{b,c,d\}$ for some $d$. The negations of the four lettered conditions are established as follows. Throughout, $a$, $b$, $c$, and $d$ will be algebraically independent over $\F_p$. \begin{compactitem} \item[$\lnot\ref{ext}$.] We have \begin{align*} (\F_p(a,b^{1/p},c),a,b,c)&\models T_1,& (\F_p(a,b^{1/p},c^{1/p}),a,b)&\models T_0, \end{align*} but if $(\F_p(a,b^{1/p},c),a,b,c)$ is a substructure of a model $(K,a,b,c)$ of $T_1$, then $K$ cannot contain $c^{1/p}$. \item[$\lnot\ref{red}$.] $T_0$ has no existentially closed models, since an element of a model that is $p$-independent from $a$ or $b$ will always have a $p$-th root in some extension. Similarly, no model of $T_1$ in which $\{a,c\}$ is not $p$-independent is existentially closed. But $T_1$ does have existentially closed models, which are just the separably closed fields of characteristic $p$ with $p$-basis $\{a,c\}$ and with an additional element $b$. \item[$\lnot\ref{AP}$.] $T_0$ does not have the Amalgamation Property, since $(\F_p(a,b^{1/p},c),a,b)$ and $(\F_p(a^{1/p},b,c,d),a,b)$ are models that do not embed in the same model over the common substructure $(\F_p(a,b,c),a,b)$, which is a model of $T_0$. \item[$\lnot\ref{ec}$.] $T_1$ is not $\forall\exists$, since, as we have already noted, models in which $\{a,c\}$ is not $p$-independent do not embed in existentially closed models. \end{compactitem} \item\label{2} For an example of the column headed by $\ref{2}$ in the table, we let $\sig_0$ and $\sig_1$ be as in $\ref{1}$; but now $T_1$ is the theory of fields of characteristic $p$ with distinguished elements $a$, $b$, and $c$ such that $\{a,c,d\}$ or $\{b,c,d\}$ is $p$-independent $\{b,c,d\}$ for some $d$. This ensures that $T_1$ has no existentially closed models, so $\ref{red}$ holds vacuously; but the other three conditions still fail. \item\label{3} $T_0$ and $T_1$ are the same theory, so $\ref{ext}$ and $\ref{red}$ hold trivially; and this theory is the theory of vector-spaces of dimension at least $2$, in the signature of vector-spaces, so it neither has the Amalgamation Property, nor is $\forall\exists$. \item\label{4} $T_1$ is $\DF_p$ with the additional requirement that the field have $p$-dimension at least $2$; and $\sig_0=\sig_{\mathrm f}$, so $T_0$ is the theory of fields of characteristic $p$ with $p$-dimension at least $2$. The latter theory has the Amalgamation Property; but the other conditions fail. Indeed, let $(\F_p(a,b),D)$ be the model of $T_1$ in which $Da=1$ and $Db=0$: then the field $\F_p(a,b)$ embeds in $\F_p(a^{1/p},b)$, which is a model of $T_0$, but $D$ does not extend to this field. Also, $T_0$ has no existentially closed models; but $T_1$ does, and indeed it has a model-companion, namely $\DCF_p$. Also $T_1$ is not $\forall\exists$, since $T_0$ is not: there is a chain of models of the latter, whose union is not a model, and we can make the structures in the chain into models of $T_1$ by adding the zero derivation. \item\label{5} $\sig_0=\sig_{\mathrm f}$, and $\sig_1=\sig_0\cup\{a\}$. $T_1$ is the theory of fields of characteristic $p$ with distinguished element $a$, which is $p$-independent from another element; so $T_0$ is (as in $\ref{4}$) the theory of fields of characteristic $p$ with $p$-dimension at least $2$. Here $T_1$ has no existentially closed models, so $\ref{red}$ holds trivially. \item\label{6} $T_0$ and $T_1$ are the same, namely the theory of fields of characteristic $p$ of positive $p$-dimension, in the signature of fields, so this theory has the Amalgamation Property, but is not $\forall\exists$. \item\label{7} $\sig_0=\sig_{\mathrm{vs}}$, $\sig_1=\sig_0\cup\{\parallel,\vec a,\vec b\}$, and $T_1$ is axiomatized by $\VSp2\cup\{\vec a\nparallel\vec b\}$, so it is $\forall\exists$. Then $T_0$ is the theory of vector-spaces of dimension at least $2$. As in Theorem~\ref{thm:vs} above, $T_1$ has a model-companion, namely the theory of vector-spaces over algebraically closed fields with basis $\{\vec a,\vec b\}$. But $T_0$ has no existentially closed models, since for all independent vectors $\vec a$ and $\vec b$ in some model, the equation $x*\vec a+y*\vec b=\vec 0$ is always soluble in some extension. Thus $\ref{red}$ fails. Then $T_0$ also does not have the Amalgamation Property, since the solutions of the given equation may satisfy $2x^2=y^2$ in one extension, but $3x^2=y^2$ in another. Similarly, $\ref{ext}$ fails, since the reduct to $\sig_0$ of a model of $T_1$ may embed in a model of $T_0$ in which $\vec a$ and $\vec b$ are parallel. \item\label{8} $\sig_0=\sig_{\mathrm{vs}}\cup\{\parallel\}$, $\sig_1=\sig_0\cup\{\vec a,\vec b\}$, and $T_1$ is axiomatized by $\VSp2$ together with \begin{equation}\label{eqn:ab} \Forall x\Forall y(x*\vec a+y*\vec b=\vec0\lto2x^2=y^2). \end{equation} Then $T_0$ is the theory of vector-spaces such that either the dimension is at least $2$, or the scalar field contains $\surd2$. As in $\ref{7}$, $T_0$ does not have the Amalgamation Property. The theory $T_1$ is $\forall\exists$. It also has the model $(\Q*\vec a\oplus\Q*\vec b,\vec a,\vec b)$, and $\Q*\vec a\oplus\Q*\vec b$ embeds in the model $\Q(\surd2,\surd3)*\vec a$ of $T_0$ when we let $\vec b=\surd3*\vec a$; but then the latter space embeds in no space in which $\vec a$ and $\vec b$ are as required by \eqref{eqn:ab}. So $\ref{ext}$ fails. Finally, $T_1$ has a model-companion, axiomatized by $\VSpst2$ together with \begin{equation*} \Exists x\Exists y(x*\vec a+y*\vec b=\vec0\land 2x^2=y^2\land x\neq0); \end{equation*} and $T_0$ has a model-companion, which is just $\VSpst2$; so $\ref{red}$ holds. \item\label{9} $T_0$ and $T_1$ are both $\VSp1$. \item\label{10} $T_1=\DF_p$, and $T_0$ is the reduct to $\sig_{\mathrm f}$, namely field-theory in characteristic~$p$. \item\label{11} $T_0$ and $T_1$ are both field-theory. \qedhere \end{asparaenum} %\end{asparaenum} \end{proof} Now let $\oDCF_0=\bigcup_{m\in\upomega}\mDCF_0$. \begin{theorem}\label{thm:oDF0} For all $m$ in $\upomega$, \begin{equation*} \mDCF_0\included\mpDCF_0. \end{equation*} Therefore $\oDF_0$ has a model-companion, which is $\oDCF_0$. This theory admits full elimination of quantifiers, is complete, and is properly stable. \end{theorem} \begin{proof} Suppose $(L,\partial_0,\dots,\partial_{m-1})$ is a model of $\mDF_0$, and $L$ has a subfield $K$ that is closed under the $\partial_i$ (where $i0\implies\tilde{\partial}_ia^{\sigma-\bm i}=a^{\sigma},\\ \label{eqn:alg} \sigma(m)>0\land a^{\sigma-\bm m}\notin K(a^{\xi}\colon\xi<\sigma-\bm m)\alg\implies a^{\sigma}\notin L(a^{\xi}\colon\xi<\sigma)\alg. \end{gather} These conditions ensure that the derivations $\tilde{\partial}_i$, if they do exist, are unique. This uniqueness of the $\tilde{\partial}_i$ would be ensured, if the field $L$ in the conclusion of \eqref{eqn:alg} were replaced with $K$. But the condition \eqref{eqn:alg}, as it is, along with \eqref{eqn:K}, \eqref{eqn:m0}, and \eqref{eqn:cond}, will ensure that, if $i0$ and $a^{\sigma}\in L$. Then by \eqref{eqn:alg} we have $a^{\sigma-\bm m}\in K(a^{\xi}\colon\xi<\sigma-\bm m)\alg$, and therefore, by \eqref{eqn:K} and \eqref{eqn:cond}, applying $\tilde{\partial}_m$ to $a^{\sigma-\bm m}$ yields $a^{\sigma}\in K(a^{\xi}\colon\xi<\sigma)$. Repeating this result, we obtain that $a^{\sigma}$ is a rational function over $K(a^{\xi}\colon\xi(m)=0)$ of certain $a^{\eta}$, where $\eta<\sigma$; and these $a^{\eta}$ are algebraically independent over $L$. Since $a^{\sigma}\in L$, we conclude $a^{\sigma}\in K(a^{\xi}\colon\xi(m)=0)$. The last observation means that, once we have constructed $K(a^{\xi}\colon\xi\in\upomega^{m+1})$ as desired, then, if $L\setminus K(a^{\xi}\colon\xi\in\upomega^{m+1})$ is nonempty, we can repeat the process, using an element of this difference in place of $a$. Thus, ultimately, we shall obtain the desired model of $\mpDF_0$ whose universe includes $L$. We shall build up $K(a^{\xi}\colon\xi\in\upomega^{m+1})$ recursively, and we shall (simultaneously) establish by induction that the desired conditions are satisfied. We shall use the ordering of the $\sigma$ in $\upomega^{m+1}$ determined by the left-lexicographic ordering of \begin{equation*} (\sigma(m),\sigma(0)+\dots+\sigma(m-1),\sigma(0),\sigma(1),\dots,\sigma(m-2)). \end{equation*} Then $\upomega^{m+1}$ has the order-type of $\upomega$ itself, and we shall have \begin{equation*} K(a^{\xi}\colon\xi\in\upomega^{m+1}) =\bigcup_{\tau\in\upomega^{m+1}}K(a^{\xi}\colon\xi<\tau). \end{equation*} When $\tau=(0,\dots,0,1)$, then, using \eqref{eqn:m0} as a definition, we have the field $K(a^{\xi}\colon\xi<\tau)$ as desired. Suppose we have this field as desired for some $\tau$ in $\upomega^{m+1}$ such that $\tau(m)>0$. In particular, for all $i$ in $m+1$, we have $\tilde{\partial}_i$ as a derivation from $K(a^{\xi}\colon\xi+\bm i<\tau)$ to $K(a^{\xi}\colon\xi<\tau)$, and the conditions \eqref{eqn:cond} and \eqref{eqn:alg} hold for all $\sigma$ such that $\sigma<\tau$. For defining $a^{\tau}$, there are two cases to consider: \begin{compactenum}[1.] \item If $a^{\tau-\bm m}$ is not algebraic over $K(a^{\xi}\colon\xi<\tau-\bm m)$, then we let $a^{\tau}$ be transcendental over $L(a^{\xi}\colon\xi<\tau)$, as required by \eqref{eqn:alg}. We are then free to define $\tilde{\partial}_ma^{\tau-\bm m}$ as $a^{\tau}$. \item If $a^{\tau-\bm m}$ is algebraic over $K(a^{\xi}\colon\xi<\tau-\bm m)$, then $\tilde{\partial}_ma^{\tau-\bm m}$ is determined as an element of $K(a^{\xi}\colon\xi<\tau)$, and we let $a^{\tau}$ be this element. \end{compactenum} We now must check that, when $i0$, we can define $\tilde{\partial}_ia^{\tau-\bm i}$ as $a^{\tau}$. Again we consider two cases. \begin{compactenum}[1.] \item Suppose $a^{\tau-\bm i}$ is algebraic over $K(a^{\xi}\colon\xi<\tau-\bm i)$. Then $\tilde{\partial}_ia^{\tau-\bm i}$ is determined as an element of $K(a^{\xi}\colon\xi<\tau)$. Thus the value of the bracket $[\tilde{\partial}_i,\tilde{\partial}_m]$ at $a^{\tau-\bm i-\bm m}$ is determined. But also, by \eqref{eqn:alg}, $a^{\tau-\bm i-\bm m}$ must be algebraic over $K(a^{\xi}\colon\xi<\tau-\bm i-\bm m)$. Since the bracket is $0$ on this field, it is $0$ at $a^{\tau-\bm i-\bm m}$ as well \cite[Lem.~4.2]{2007arXiv0708.2769P}. \item If $a^{\tau-\bm i}$ is transcendental over $K(a^{\xi}\colon\xi<\tau-\bm i)$, then since we are given $\tilde{\partial}_i$ as a derivation whose domain is this field, we are free to define $\tilde{\partial}_ia^{\tau-\bm i}$ as $a^{\tau}$. \end{compactenum} Thus we have obtained $K(a^{\xi}\colon\xi\leq\tau)$ as desired. Therefore $\oDF_0$ has the model-companion $\oDCF_0$. As noted, $\oDCF_0$ inherits quantifier-elimination, completeness, and stability from the $\mDCF_0$, which have these properties \cite{MR2001h:03066}. Although each $\mDCF_0$ is actually $\upomega$-stable, $\oDCF_0$ is not even superstable, since if $A$ is a set of constants (in the sense that all of their derivatives are $0$), then as $\sigma$ ranges over $A^{\upomega}$, the sets $\{\partial_mx=\sigma(m)\colon m\in\upomega\}$ belong to distinct complete types. \end{proof} We may note that, in the foregoing proof, we cannot use Condition $\ref{ext}$ of Theorem~\ref{thm:cond} in the stronger form in which the structure $\str C$ is required to be a mere \emph{expansion} to $\sig_1$ of $\str B$: \begin{theorem} If $m>0$, there is a model of $\mDF_0$ that does not expand to a model of $\mpDF_0$. \end{theorem} \begin{proof} We generalize the example of \cite{MR96g:35006} repeated in \cite[Ex.~1.2, p.~927]{MR2000487}. Suppose $K$ is a pure transcendental extension $\Q(a^{\sigma}\colon\sigma\in\upomega^{m+1})$ of $\Q$, and $\partial_ia^{\sigma}=a^{\sigma+\bm i}$. Let $L$ be the pure transcendental extension $K(b^{\tau}\colon\tau\in\upomega^{m-1})$ of $K$, and if $i0)$. However, $\partial_{m-1}x$ is algebraic over this field whenever $x$ is algebraic over $K$. Thus all of the $\partial_m{}^kb^{\tau}$ are algebraically independent over $K$; in particular, when $k>0$, they do not belong to $L$. \end{proof} Finally, the union of a chain of non-companionable theories may be companionable: \begin{theorem} In the signature $\{f\}\cup\{c_k\colon k\in\upomega\}$, where $f$ is a singulary operation-symbol and the $c_k$ are constant-symbols, let $T_0$ be axiomatized by the sentences \begin{equation*} \Forall x\Forall y(fx=fy\lto x=y) \end{equation*} and, for each $k$ in $\upomega$, %\begin{gather*} \begin{align*} \Forall x(f^{k+1}x&\neq x),& \Forall x(fx=c_k&\lto x=c_{k+1}),& fc_{k+2}=c_{k+1}&\lto fc_{k+1}=c_k. \end{align*} %\end{gather*} For each $n$ in $\upomega$, let $T_{n+1}$ be axiomatized by \begin{equation*} T_n\cup\{fc_{n+1}=c_n\}. %\land fc_n=c_{n-1}\land\dots\land fc_1=c_0\}. \end{equation*} Then \begin{compactenum} \item each $T_n$ is universally axiomatized, and \emph{a fortiori} $\forall\exists$, so it does have existentially closed models; \item each $T_n$ has the Amalgamation Property; \item every existentially closed model of $T_{n+1}$ is an existentially closed model of~$T_n$; \item no $T_n$ is companionable; \item $\bigcup_{n\in\upomega}T_n$ is companionable. \end{compactenum} \end{theorem} \begin{proof} Let $\str A_m$ be the model of $T_0$ with universe $\upomega\times\upomega$ such that \begin{align*} f^{\str A_m}(k,\ell)&=(k,\ell+1),& c_k{}^{\str A_m}&= \begin{cases} (k-m,0),&\text{ if }k>m,\\ (0,m-k),&\text{ if }k\leq m. \end{cases} \end{align*} Let $\str A_{\upomega}$ be the model of $T_0$ with universe $\Z$ such that \begin{align*} f^{\str A_{\upomega}}k&=k+1,& c_k{}^{\str A_{\upomega}}&=-k. \end{align*} Then $\str A_m$ is a model of each $T_k$ such that $k\leq m$; and $\str A_{\upomega}$ is a model of each $T_k$. Moreover, each model of $T_k$ consists of a copy of some $\str A_{\beta}$ such that $k\leq\beta\leq\upomega$, along with some (or no) disjoint copies of $\upomega$ and $\Z$ in which $f$ is interpreted as $x\mapsto x+1$. Conversely, every structure of this form is a model of $T_k$. The $\beta$ such that $\str A_{\beta}$ embeds in a given model of $T_k$ is uniquely determined by that model. Consequently $T_k$ has the Amalgamation Property. Also, a model of $T_k$ is an existentially closed model if and only if includes no copies of $\upomega$ (outside the embedded $\str A_{\beta}$): This establishes that every existentially closed model of $T_{k+1}$ is an existentially closed model of $T_k$. The existentially closed models of $T_k$ are those models that omit the type $\{\Forall yfy\neq x\}\cup\{x\neq c_j\colon j\in\upomega\}$. 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